lilt. J. Jlech. Sci. Pergamon Press Ltd. 1962. \'01. 4, pp. 13i-146. Printed in Great Britain

CREEP BENDING OF COMPRESSIBLE PLATES*

SUARAD A. PATEL and B. VENKATRAl\IANPolytechnic Institute of Brooklyn, New York

(Receit"cd 24 May 1961)

Summary-A nonlinear moment-curvature relation is developed for the creep-bendinganalysis of thin plates made of compressible materials. This relation is used to analyse twospecific annular plates: an annular plate simply supported along the outer edge and freeat the inner edge, with uniform radial moments applied along the supported circumference;an annular plate clamped along the inner edge and free at the outer edge, with uniformradial moments applied along the free circumference. Solutions for moment distributionsand deflexions are presented for both problems.

1. INTRODUCTION

FOR theoretical investigations of creep in structures subject to combinedstresses an extremely useful basis is the general stress-strain relations ofPrager", formulated for incompressible isotropic materials in the strain­hardening range. It was shown by Hoff" that this formulation could be readilyadapted for creep analyses by means of the elastic analogue. Further, anelementary device for the inclusion of compressibility effects in the creep ofmaterials was presented in Ref. 3. It was shown that the effect of compressi­bility could be included by modifying the definition of stress deviations inPrager's! stress-strain relations. This modified formulation is used in thepresent paper to investigate the creep bending of circular plates made ofcompressive materials.

Section 2 begins with a discussion of the assumptions under which it isshown that the modified form of the stress-strain law is the elastic analogue ofthe creep law. This is used as a basis in the derivation of a moment-curvaturerelation for the analysis of plates. These relations are then used to analysetwo specific annular plates: (1) An annular plate simply supported along theouter edge and free at the inner edge, with uniform radial moments appliedalong the supported circumference. (2) An annular plate clamped along theinner edge and free at the outer edge, with uniform radial moments appliedalong the free circumference. Solutions for moments and deflexions are obtainedin both cases. The conclusions are summarized in a final section.

2. l\IOMENT-CURVATURE RELATIONS

The consequence of the assumption of incompressibility in Prager's stress­strain formulation is that the deformations are produced only by tho deviationof stresses from a hydrostatic state of stress. Since a compressible material

* The results presented in this paper were obtained in course of research sponsored by theAir Force Office of Scientific Research of the Air Research and Development Command underContractAF 49(638)-302.

137

138 SUARAD A. PATEL and B. VENKATRA~IAN

would deform even under hydrostatic stress, its deformations cannot beexpressed simply as a function of stress deviations from this state. However, amodified definition of stress deviations was proposed in Ref. 3 to permit theinclusion of compressibility effects.. A .relatively simple form of Prager's lawwas used in this reference to express the strain as

6il = CJ2msij (2.1)

with the stipulation that the usual definition of the stress deviations Sil bealtered to read

sil = Uil - (Xukk oil (2.2)

In these equations, C and m are material constants, J 2 = !Sil "u is the secondinvariant of the "modified deviator" sil' uil is the stress tensor, oil is theKronecker delta and ex is the compressibility constant.* When ex = 1/3, themodified stress-deviation expression (2:2) corresponds to an incompressiblematerial as given in Ref. 1.

Ifthe strain is sufficiently small for the linear strain-displacements equationsto be used, Hoff2 has shown that the nonlinear stress-strain relation (2.1) is theelastic analogue of the creep law

6i} = C.1rsijtl/Q (2.3)

where q is a creep constant and t represents time. Now, the creep analysis of astructure under combined stresses may be based on equation (2.3) providedelastio deformations are negligibly smaller than creep deformations and thestructure is at a uniform temperature, so that no thermal stresses exist. Subjectto these assumptions, the creep problem of a structure governed by the law(2.3) is mathematically equivalent to the nonlinear elastic problem of the samestructure governed by the law (2.1). Therefore, the present consideration ofthe creep bending of thin plates will rest upon the stress-strain relations (2.1).

Investigations of the bending of thin plates out of their plane arc usuallybased on derived moment-curvature relations. This procedure is followed toderive these relations from the stress-strain law (2.1). As a first step, thedefinition J 2 = lSi} Sil is used together with equation (2.1.) to express theinvariant J 2 in terms of the strain tensor 6il as

J2 = (6i} 6ij/2C2)l/(2m+I1 (2.4)

Use of this expression in equation (2.1) shows that

_ 6ijSij - CI/(2mHI (6.. 6 .. /2)m/(2m+11 (2.5)

l] l]

which has a readily usable form in deriving the moment-curvature relationsfor thin plates. The small-deflexion theory of plates is assumed to be validin the following development.

The problem of bending of thin plates out of their plane is one of generalizedplane stress. Therefore, the normal stress and shear stresses in the direction ofthe co-ordinate perpendicular to the plane of the plate are identically zero. If it

* It was shown in Ref. 3 that ex must satisfy the condition 0":; ex":; 1/3.

Creep bending of compressiblo plates 139

(2.7)

(2.6)

is assumed that this direction is represented by the third co-ordinate, the stressterms 'with the number 3in their subscripts vanish identically. In that case,using the modified definition of sij (equation 2.2), a simple transformation ofequation (2.5) yields the stress tensor

Eij+aEkI~Sii/(1-2a) •. 1?eJ·· = , 't,J = ,~

t] {(a/2m) [Eij Eij +{af(1- 2a)}2<«EjJ]mp/(2m+!1

If z denotes the distance in the perpendicular (third) direction of a genericpoint in the plate from the neutral plane, Eij = ZKij is the strain-curvaturerelation, and hence equation (2.6) assumes the form

_ [Kij +aKkk 8d(l- 2a)] zlf(2m+!1eJ·· - -;-;-;~~~--"'-'7-~'------:::-~'-----~;-;-;-;~:-=

t] {(a/2m) [Kij Kij+{a/(1- :?a)}2 <«Kjj)'U}1!(211l+!l

The moment tensor per unit length of the plate is given by the definition

(2.9)

(2.8)

where

f+h' 211Iij = zeJij dz

-ta«

where It is the uniform thickness of the plate. Equations (2.7) and (2.8) ma.ynow be combined and simplified to yield the relation

111.. = K Kij+aKkk 8ijf(1- 21X)t] {![Kij Kij+{a/(1- 2a)}2<«Kjj]}lIl/(211l+!1

(2.10)4m +2 [(h/2)1m+3] 1/(2m+!1

J(=--4m+3 a

It can be shown, with the use of equation (2.9), that the invariant

~ [KijKij+ (1:2ar KiiKjj] = {[.ilIij.ilIij -a(2 - 31X).i11u11l;i]/2f{2Fm+! (2.11)

Introduction of this relation into equation (2.9) and simplificationyield thecurvature

= [MijMij-a(2- 3a)MuMjJ]m (11Iii-a.ilIkkSij) (2.12)Kii 2f{2 J(

This moment-curvature relation is used as a basis for the analysis of particularproblems in the next section.

3. ILLUSTRATIVE EXAMPLES

The relatively simple problems of annular plate subject to uniform momentsalong the edges (Fig. 1) prove convenient in illustrating the application of theabove relations. The radial symmetry of these problems requires that theradial and tangential co-ordinates Rand 8 correspond to principal directions.Equation (2.12) therefore yields the principal curvatures

= -lV" = [(1-m(lII;+1I[~)-2f3.Mr.Mo]m [(1-a)1II,.-WlIo]K r 2](2 tc

KfJ = - ~' = [(1-f3)(111;~::)- 2f31lIr1lIoJm [{1-a)~O-a11I,.] (3.1)

140 SU.illAD A. PATEL and B. VEXKATRA)IA.."

where (3 = 0:(2- 30:), TV is the deflexion, and the primes denote differentiationwith respect to R. Further, these plates (Fig. 1) are governed by the equi­librium equation

(3.2)

Equations (3.1) and (3.2) determine the moments M,. and Mo and thedeflexion TV.

(0)

J!-----1J M

r

o

IrI

Jij1----- Ro ---ri

(b)

FIG. 1. Annular plates.

Elimination of TV between the two equations (3.1) yields the momentrelation

(3.4)

(3.3)m[(1- (3) (Jl1~ +Jl1~) - 2{3J1I,.MoJ' _ (M,. - ..Mo)- R[(l- 0:) .illo- o:M;]

[(1- (3)(J1l~+111~) - 2f3.L1I,..I1Io' - R[(1- 0:) No - a:.L1I,.]

Equations (3.2) and (3.3) may now be solved simultaneously to obtain themoment distributions .L14 and .Mo. To this end, defining a new variable

RJ1l; = Nr/y

the equilibrium equation (3.2) is reduced to

.<:lIe = .I1I,.(1 + I/y)

Use of these relations simplifies equation (3.3) to the form

dR (2m+ 1)+[2ma+(m+ l)b]y+b(ma+ 1)y2 dIf = (1 +by+by2)[(2m+ 1) +2(ma+ I)y] y

while equation (3.4) may be transposed to read

d.I1Ir . 1 dR11/,. = !/If

The constants a and b in equation (3.6) have the following values:

(3.5)

(3.6)

(3.7)

a = (1-20:)/(1-0:), b = 2(1-2f3)/(1-f3) (3.8)

Creep bending of. compressible plates 141

(3.9)

As a first step in the solution of the two problems considered, equations (3.6)and (3.7) are integrated to yield

['Jy+l ]AR = f(y) = (y+n/q)P (1 +by+by2) (l- 2P l/! exp k1 tan-1 ~(~/b-l)

B.LlI, = g(y) = y(y+n/q)-pq/n (l +by + by2)(pq/n-1l/2 exp [k 2tan-1 ~(~~b+-\)] (3.10)

Hereb/2-2

11 = b(q-n)-q2' q = (n-l)a+2

k - 4(1-pq/n)-b(1-2p) l: _ b(n+l-pq)-(n+4-2p q2/n )1 - 2b~(4/b-l) , '2 - 21lb~(4/b-l)

and A and B are arbitrary constants. Equations (3.9) and (3.10) are now usedin the investigation of the two specified problems.

The annular plate on simple supports shown in Fig. l(a) is considered first.Since the moment at the inner edge-11f" = 0, equations (3.4) and (3.5) show that11fo, = RiM;,. If 11fo, = RiM;, =P 0, it follows from -these equations that thecorresponding values of Yi at the inner edge must be zero. The final solutionwill verify that 11fo, =P 0, and hence that Yi = O. Theboundary condition Yo atthe outer edge is obtained as follows. The corresponding boundary valuesRo and Yo as well as Ri and Yi are, respectively, substituted in equation (3.9).A division of the resulting equations leads to

(3.11)

The solution of this equation for given values of Ro/Ri yields Yo'Having determined Yo' equations (3.9), (3.10) and (3.5) are nondimen­

sionalized to readr = R/Ro = f(y)/f(yo)

m, = l1J,./.Mr• = g(y)/g(yo)

mo = Mo/Mr• = (1 + l/y)g(y)fg(yo)

(3.12)

(3.13)

(3.14)

These equations are used to evaluate 112r and 1120 distributions along the radius r.The plate deflexion is obtained by integration of the second of equations (3.1).Hence, in nondimensional form,

TV J'[ (1)] [(1-f3)(112~+112~)-2f3mr1120]1lldtv = Rij(.Llf

r) K )2Tn H = 1 0:1Ilr- -0: 1120 2 r r

(3.15)where the integral is evaluated numerically.

The annular plate _with fixed 'inner edge shown in Fig. l(b) is consideredas a second example. Since the slope of the plate at the inner edge is zero, itfollows from the second of equations (3.1) that .Mo, = O:.Llf,J( 1 - 0:). Substitutionof this relation in equation (3.5) yields Yi =: (1- 0:)/(20: - 1). The value of Yo isthen determined, for given values of Ro/Ri , by equation (3.11). Substitutingfor y and Yo in equations (3.12), (3.13) and (3.14), the moment distributions are

10

142 SIIARAD A. PATEL and B. VE~"KATRAlIIAN

2-S

2-4

2-0

1-8

,,2

'-0

0-6

(}2

no'for olla l-

I

\

\

\

I \

I

I

I

1

,~- -

//

I

,I

2·2

2-0

1·8

/·6

'-4

\-0

os

0·6

04

0·2

n=7

00 ...

...1/6 -1/3i> ... ~~

...

a0

\ 1;6-;' 0

17'~~:/3~e~C-"

~ /

r-f/- ~e6 /1/3

VIht'--O -

1/ [S.ye1/3 -

III

o (}2 (}4 0-6 O'S '-0 o 02 04 0-6 0-8

r

,,0

2·2

2·0

"8

1·4

\-0

(}8

(}6

0-4

0·2

a n030

\

1/6 <,1/3-~\

"t,: ...... ...

00

16

1/3

a

=~~~ ;/

~ r--rf /r :--.:: 16

13

IfJ::-o0

-I- ..... ~~ -

III

2-4

2·2

2·0

1·8

1·6

1·4

E! 1.2

E

\-0

os

(}2

n=ro

00 ...

.........., -, ,- ...1/6

1/3l/

00

\

I/~ ...\

1/3 - I":"~

~~ Vf--P-~~6 ,/

1/3

1/,>g

f-'l- I'-.. 1/6

13 f--I

/.

o Q-2 0·4 (}6 Q-S 1-0 o 0-2 Q-4 0-6 (}8 i-o

FIG. 2. Moment distributions for simply supported plates.m,me - ----'

Creep bending of compressible plates 143

n=1

1·8H+-+--+--+-j-t--+-+-+----t

1/6

1'61-*,f--+--+--+ ao

\1/6 \ \

-- -

I,

113 \l\

I '

II I

i:' / /4- 116 c-- - 116Y--1f--I--I--t--iI

\'2

E[ 1·0

E

(}8

0-6

0'4

0·2I,

I0-21-+'-+--t--t--t-I-i't--+--+--t--i

1

If

E. ~::::..1'0

E I ~~f~--j-+--+--+--t--t---l0-8 -!, 1~1/6 -jc-t--+---t::;~=t-i

[, 0

0 1

n=7

a0

a1/6 0

1/6 \!l3 1\ r\.1/3

1""- '-.:~ .......p:::~~/'

I -- - -II /' -, -I

a a,

/,

13,

1/3VI

,II

)/6 1/ 1I1/6 /

/'0 0

(}2 0·4 0-6 0 8

n=co

a

II 0 a0

II" \.13 1/6 e-, -,1/3

" ~ ::--..~ -- '-::

17 .::;: -- '-l /1 ,/

f-Jf, a - f- a u /I'

1/3 1/3 !/Ii It

" III

t llS 1/6 I

II

1'0 0

\'0

1'0

0-6 0-80'

0-2 0-4

0-2 0·4 (}6 0'8

I 0

I'S

1'8

1·4

o2-0

o

0·2

0-6

0-4

(}8

E. \'0

E

10

1-0(}4 (}S 0'80·2o

1-8

1·6

1·4

1'2

0-4

o2-0

0-8

(}S

E. 1-0

E

r

FIG. 3. Moment distributions for clamped plates.1n r ---­

1no---

144 SIIARAD A. PATEL and B. VE~KATRA~L\S

~ I kI"'~

a )'21/3

"~ 1/6 / V~~ /

I n'l(

o~ ;::-"-- -~~

""'"r-r- f-- 1-~~~I'""--- V 1/3 f-r-- 1- 1/6

0 f-

n:(

I

o

w

~ ~\\ "- 1 v '/a

\ \ ~I/~,

/11i\ 1~6 /

\ /1

\ /\

~ I\·0

n:32

o<;-- ~ .....-V

<, :~~r- .-- "\ /3 f-

r-- - /60 I-

n;3

1

o

w

o

IY ...;;;;:: ,:""-'/h .... / -/\ '116 V

\ /~ /'-0 ,

1 n:7

o

( (o

1"'- \\ /<, ~~

'- .-- ~/3 -- - /6

10 -

A:7

I

o

w

FIG. 4. Deflexions for simply supported plates.

Creep bending of compressibl e plates 145

evaluated. Finally, integration of the first of equations (3.1) yields thodimensionless deflexion

TV f.rf.r [(I-f3)(m~+m~)-2f3mrmoJmw = Bij(.J.1J

r,/K)2In+I = r/ r/[o:mo-(I-O:)1nr] 2 drdr

(3.16)

w

w

0 ·6

0·4

0 '2

o

n a 3

1\ a 1-~,~\. 1/3"

r-, .

f\. 1/1'"r-

I"-.... Vt--l........ i'... L'-.. / ~ ,..... Po-

0·4

o

0 '4

w 0 ·2

o

n a I

r- r- a

~~6.I

1/3~~

I n=3

r-t-I"

~~;/

~ b::::::

w

1·0

08

0 6

04

0 ·2

n a 7

1\ /\ 1/

\ /, II

1\ /\ ~g t--1\ J 116

/113~-

t-. I'... ~

o

0 ·4

w 0·2

I n a 7

~U"~~6 L

1/3 Vo

F IG. 5. Deflexions for clamped pl at es.

Th e moments m, and mo and the deflexion w are shown plotted in Figs. 2,3, 4 and 5. The specific valu es chosen for the compressibili ty constant 0:, thecreep-law index n , and the radii ra t io Ro/Ri arc 0: = 0, I/G, 1/3; n = 1,3,7,00;B olRi = 2, 20.

4. CONCL U D I N G RE nIARKS

A nonlinear moment-curvature relation has been developed for the creep­bending analysis of thin plates made of compressible materials. This relationhas becn used to obtain solutions of two specific annular-plate problems. It is

146 SUARAD A. PATEL and B. VENKATRA]IAN

hoped that the derivation presented would prove successful in the solution ofother plate problems under more complex loading conditions. It should bementioned, however, that the basis for the moment-curvature relation developedis comparatively recent and any theoretical prediction must be verified byadequate test results.

Acknowledgement-The authors wish to thank Miss Arlene Arons for carrying out thenumerical calculations during this research.

REFERENCES

1. W. PRAGER, J. Appl. Phys. 16, 21, 837 (1945).2. N. J. HOFF, AGARD Report No. 52 (1956).3. S. A. PATEL, B. VENKATRAlIIAN and w. P. VAFAKOS, Int. J. 11Iech. Sci. 2,1 (1960).